Functional Analysis

Semilocal convergence of a continuation method with Hölder continuous second derivative in Banach spaces

by prashanth maroju

M.Prashanth and D.K.Gupta

In this paper, the semilocal convergence of a continuation method combining the
Chebyshev method and the convex... more In this paper, the semilocal convergence of a continuation method combining the
Chebyshev method and the convex acceleration of Newton’s method used for solving
nonlinear equations in Banach spaces is established by using recurrence relations under
the assumption that the second Frëchet derivative satisfies the Hölder continuity condition.
This condition is mild and works for problems in which the second Frëchet derivative fails
to satisfy Lipschitz continuity condition. A new family of recurrence relations are defined
based on two constants which depend on the operator. The existence and uniqueness
regions along with a closed form of the error bounds in terms of a real parameter α ∈ [0, 1]
for the solution x∗ is given. Two numerical examples are worked out to demonstrate
the efficacy of our approach. On comparing the existence and uniqueness regions for the
solution obtained by our analysis with those obtained by using majorizing sequences under
Hölder continuity condition on F ′′ , it is found that our analysis gives improved results.
Further, we have observed that for particular values of the α, our analysis reduces to those
for the Chebyshev method (α = 0) and the convex acceleration of Newton’s method
(α = 1) respectively with improved results.

Comparing the impact of the OO-DFD and the Use Case methods for modeling functional requirements on comprehension and quality of models - a controlled experiment

by Peretz Shoval

Authors: M. Dahan, P. Shoval, A. Sturm (2012)
Journal: Requirements Engineering. (in press)

Users' requirements of an information system are modeled in the analysis phase of the development process. The... more Users' requirements of an information system are modeled in the analysis phase of the development process. The requirements can be modeled with various modeling methods. In this study we compare two alternative methods for modeling the functional requirements: one is the UML Use Case (UC) model; the other is OO-DFD transactions (Object-Oriented DFD is a variant of DFD that includes data classes rather than "traditional" data stores). Each of these modeling methods consists of diagrams accompanied with narrative, semi-structured descriptions explaining their details. We conducted a controlled experiment that compared the comprehension of the two models (i.e., the diagrams and their descriptions) of a certain system, and the quality of models created for a certain system with each of the two modeling methods. The main results of the experiment are that models created with the UC method are of better quality than models created with the OO-DFD transactions method because the former are simpler and less detailed; creating highly-detailed models are error-prone. Interestingly, in spite of the difference in the level of detail and structure, the experiment reveals no significant difference in comprehension of models of the two methods. The results call for improvement of the modeling methods in a way that considers the advantages of each of them; and thus we propose an improved method sketch that we call Enhanced Use Case (EUC), which will be evaluated in future work.

A Novel Dynamic Impact Approach (DIA) for Functional Analysis of Time-Course Omics Studies: Validation Using the Bovine Mammary Transcriptome

by Massimo Bionaz

PLoS ONE, 2012

The overrepresented approach (ORA) is the most widely-accepted method for functional analysis of microarray datasets.... more The overrepresented approach (ORA) is the most widely-accepted method for functional analysis of microarray datasets. The ORA is computationally-efficient and robust; however, it suffers from the inability of comparing results from multiple gene lists particularly with time-course experiments or those involving multiple treatments. To overcome such limitation a novel method termed Dynamic Impact Approach (DIA) is proposed. The DIA provides an estimate of the biological impact of the experimental conditions and the direction of the impact. The impact is obtained by combining the proportion of differentially expressed genes (DEG) with the log2 mean fold change and mean –log P-value of genes associated with the biological term. The direction of the impact is calculated as the difference of the impact of up-regulated DEG and down-regulated DEG associated with the biological term. The DIA was validated using microarray data from a time-course experiment of bovine mammary gland across the lactation cycle. Several annotation databases were analyzed with DIA and compared to the same analysis performed by the ORA. The DIA highlighted that during lactation both BTA6 and BTA14 were the most impacted chromosomes; among Uniprot tissues those related with lactating mammary gland were the most positively-impacted; within KEGG pathways ‘Galactose metabolism’ and several metabolism categories related to lipid synthesis were among the most impacted and induced; within Gene Ontology “lactose biosynthesis” among Biological processes and “Lactose synthase activity” and “Stearoyl-CoA 9-desaturase activity” among Molecular processes were the most impacted and induced. With the exception of the terms ‘Milk’, ‘Milk protein’ and ‘Mammary gland’ among Uniprot tissues and SP_PIR_Keyword, the use of ORA failed to capture as significantly-enriched (i.e., biologically relevant) any term known to be associated with lactating mammary gland. Results indicate the DIA is a biologically-sound approach for analysis of time-course experiments. This tool represents an alternative to ORA for functional analysis.

Functional Analysis and treatment of the delusional statements of a man with multiple disabilities: A four-year follow-up

by Robert Travis

Although delusional statements in people with intellectual disabilities and traumatic brain injury can be treated... more Although delusional statements in people with intellectual disabilities and traumatic brain injury can be treated using behavioral interventions, none have demonstrated long-term treatment
effects. In the current study, a functional analysis demonstrated that delusional statements were maintained by attention. Differential reinforcement of alternative behavior and extinction of delusional statements resulted in near elimination of delusional statements and an increase in nondelusional statements. Follow-up at 6 months, 1, 2, and 4 years indicated that treatment gains were maintained with continued staff training.

Data modeling or functional analysis: which comes first? an experimental comparison

by Peretz Shoval

Authors: P. Shoval, J. Kabeli
Journal: Communications of the AIS (2005), Vol. 16, Article 42.

The software analysis process consists of two main activities: data modeling and functional modeling. While... more The software analysis process consists of two main activities: data modeling and functional modeling. While traditional development methodologies usually emphasize functional modeling via dataflow diagrams (DFDs), object-oriented (OO) methodologies emphasize data modeling via class diagrams. UML includes techniques for both data and functional modeling which are used in different methodologies in different ways and orders. This article is concerned with the ordering of modeling activities in the analysis stage. The main issue we address is whether it is better to create a functional model first and then a data model, or vice versa. We conduct a comparative experiment in which the two opposing orders are examined. We use the FOOM methodology as a platform for the experiment as it enables the creation of both a data model (a class diagram) and a functional model (hierarchical OO-DFDs), which are synchronized. The results of the experiment show that an analysis process that begins with data modeling provides better specifications than one that begins with functional modeling.

Refuge or dwelling place? The MBA fortification wall of Roca (Lecce, Italy): the spatial and functional analysis of Postern C

by Teodoro Scarano

published in: Rivista di Scienze Preistoriche LXI 2011, 95-122

The Bronze age site of Roca is a long-lived coastal settlement
ever protected by a large fortification wall,... more The Bronze age site of Roca is a long-lived coastal settlement
ever protected by a large fortification wall, which was repeatedly destroyed, rebuilt and restored over the centuries between the Middle and Final Bronze Ages. The analysis of the remains of the Late Apennine (MBA) fortifications have revealed a plan of the enceinte with a monumental main entrance and at least five postern gates. The defensive wall was destroyed by a great fire, probably subsequent to a siege, which caused the collapse
of the stonework structures, and burned down the settlement almost completely. The excavation of the rubble filling from the posterns’ long corridors and the Monumental Gate showed that the original contents of these spaces were sealed in their functional situation as it was shortly before the destruction. The most extraordinary discovery came from Postern C: several vessels were grouped in three different areas. In a fourth zone, at the western end of the corridor, another group of vessels was associated to seven complete human skeletons, two adults, a juvenile and four children. Apparently, during the siege the members of this group had taken refuge within the corridor and finally tried to hide behind a heap of vessels. Unfortunately, they all died from asphyxia caused by the fire set to the fortification wall. The archaeological record from Postern C provides a unique possibility to correlate the quantitative, qualitative and spatial data relative to the artifacts assemblages, with the human group which had brought in those objects in order to survive in that space, had planned their spatial distribution, and had used them.

Functional Analysis and Schopenhauer's Theory of the Will-Draft

by Star Maker

Traces et fonction: les gestes retrouvés

by Hugues Plisson

ANDERSON P. C., BEYRIES S., OTTE M. et PLISSON H. 1993. Traces et fonctions : les gestes retrouvés, ERAUL, CRA du CNRS, Etudes et recherches archéologiques de l'Université de Liège, Liège, 542 p.

Un burin ne sert pas à buriner mais en burinant

by Hugues Plisson

PLISSON H. 2006. Un burin ne sert pas à buriner mais en burinant... In: Burins préhistoriques : formes, fonctionnements, fonctions. / Luxembourg: MNHA, 2006 (ArchéoLogiques 2), p.23-33.

The burin belongs to the Upper Palaeolithic tool kit. It coincides with the emergence and development of bone... more The burin belongs to the Upper Palaeolithic tool kit. It coincides with the emergence and development of bone technology, which goes along with the generalisation of projectile points, and of art. It has been seen as technically linked to these three domains which are emblematic of Homo sapiens sapiens. Technological and functional studies have shown that Upper Palaeolithic is also distinguishable from previous times by an increasing complexity of technical systems, with more specialised structures whose finalities we can better understand. It is the reason why the main categories of tools distinguished by typology match with real functional entities. However, the burin escapes this fine discipline, and the archaeological assemblages which could validate the functional assumption linked to its typological denomination are not as numerous as we believed. On the contrary, use-wear analysis, along with technological studies, reveal a surprising functional diversity.

Caracterización funcional de los microlitos geométricos. El caso del Valle del Ebro

by Rafael Domingo Martínez

Published in "El Mesolítico Geométrico en la Península Ibérica"

This article deals with the functional analysis of more than 700 geometric microliths from the Ebro Valley, found in... more This article deals with the functional analysis of more than 700 geometric microliths from the Ebro Valley, found in sites from Castilla y León, Navarra and Aragón. The main part of those which showed use wear (about 35%) were employed as projectile points, but we perceived also other utilisations such as the skin work or cutting of fresh vegetables. We have established also the orientation of the projectile points hafting, strongly related with their morphology.

Unconstrained Optimization Techniques for the Acceleration of Alternating Projection Methods

by Luis Manuel Hernandez Ramos

Co-authored with R. Escalante and M. Raydan

Alternating projection methods have been extensively used to find the closest point, to a
given point, in the... more Alternating projection methods have been extensively used to find the closest point, to a
given point, in the intersection of several given sets that belong to a Hilbert space. One of
the characteristics of these schemes is the slow convergence that can be observed in practical
applications. To overcome this difficulty, several techniques, based on different ideas, have
been developed to accelerate their convergence. Recently, a successful acceleration scheme was
developed specially for Cimmino’s method when applied to the solution of large-scale saddle point
problems. This specialized acceleration scheme is based on the use of the well-known conjugate
gradient method for minimizing a related convex quadratic map. In this work, we extend and
further analyze this optimization approach for several alternating projection methods on different
scenarios. In particular, we include a specialized analysis and treatment for the acceleration of
von Neumann-Halperin’s method and Cimmino’s method on subspaces, and Kaczmarz method
on linear varieties. For some specific applications we illustrate the advantages of our acceleration
schemes with encouraging numerical experiments.

EXTRAPOLATION ON Lp,∞(µ)

by Pedro Tradacete

We solve the extrapolation problem concerning bounded operators on weak-Lp spaces; that is, we give end-point... more We solve the extrapolation problem concerning bounded operators on weak-Lp spaces; that is, we give end-point estimates for sublinear operators T such that T is bounded on weak-Lp with constant less than or equal to 1/(p-1)^m. Applications to the Hardy-Littlewood maximal operator, the Hilbert transform and composition of operators are also given.

SOME PROPERTIES OF STRICTLY SINGULAR OPERATORS ON BANACH LATTICES

by Pedro Tradacete

Actas del Colloquium del Departamento de Análisis Matemático (2009/2010).

Several results obtained during the author's Ph.D. Thesis are presented. In particular, domination results (in... more Several results obtained during the author's Ph.D. Thesis are presented. In particular, domination results (in Dodds-Fremlin sense) for the ideal of strictly singular operators will be given. Moreover, the connections between strictly singular and the classes of AM-compact, l2-singular and disjointly strictly singular are studied. As an application we obtain existence
of invariant subspaces for positive strictly singular operators. On a dierent direction, results on compact powers of strictly singular operators are also presented extending a theorem of V. Milman. Finally, we study when a c0-singular or l1-singular operator can be extended to an operator between vector valued lattices preserving its singularity properties.

Invariant subspaces of positive strictly singular operators on Banach lattices

by Pedro Tradacete

Published in 'Journal of Mathematical Analysis and Applications' 343 (2008), 743-751.

It is shown that every positive strictly singular operator T on a Banach lattice satisfying certain conditions is... more It is shown that every positive strictly singular operator T on a Banach lattice satisfying certain conditions is AM-compact and has invariant subspaces. Moreover, every positive operator commuting with T has an invariant subspace. It is also proved that on such spaces the product of a disjointly strictly singular and a regular AM-compact operator is strictly singular. Finally, we prove that on these spaces the known invariant subspace results for compact-friendly operators can be extended to strictly singular-friendly operators.

Powers of operators dominated by strictly singular operators

by Pedro Tradacete

Published in 'The Quarterly Journal of Mathematics' 59 (2008), 321-334.

It is proved that every positive operator R on a Banach lattice E dominated by a strictly singular operator T... more It is proved that every positive operator R on a Banach lattice E dominated by a strictly singular operator T satisfies that the fourth power R^4 is strictly singular. Moreover, if E is order continuous then the square R^2 is already strictly singular.

Strictly singular operators on Lp spaces and interpolation

by Pedro Tradacete

Published in 'Proceedings of the American Mathematical Society' 138 (2010), 675-686

We study the class of strictly singular non-compact operators on Lp spaces. This allows us to obtain interpolation... more We study the class of strictly singular non-compact operators on Lp spaces. This allows us to obtain interpolation results for strictly singular operators on Lp spaces. Given p < q, it is shown that an operator T bounded on Lp and Lq which is strictly singular on Lr for some r between p and q, then it is compact on Ls for every p < s < q.

Interpolation of Banach Lattices and Factorization of p-Convex and q-Concave Operators

by Pedro Tradacete

Published in 'Integral Equations and Operator Theory' 66 (2010), 79-112.

We extend a result of Sestakov to compare the complex Interpolation method with Calderón-Lozanovskii's construction,... more We extend a result of Sestakov to compare the complex Interpolation method with Calderón-Lozanovskii's construction, in the context of abstract Banach lattices. This allows us to prove that an operator between Banach lattices which is p-convex and q-concave, has a factorization whose factors have nice convexity and concavity properties.

Subspace structure of Lorentz Lp, q spaces and strictly singular operators

by Pedro Tradacete

Published in 'Journal of Mathematical Analysis and Applications' 367 (2010), 98-107.

We study subspaces of Lorentz Lpq spaces and provide an easy-to-check characterization of strictly singular operators... more We study subspaces of Lorentz Lpq spaces and provide an easy-to-check characterization of strictly singular operators defined on these spaces. As an application we obtain stability
under duality for the class of strictly singular operators on Lpq spaces, extending a theorem of L. Weis for operators on Lp spaces.

El problema del subespacio invariante en espacio de Banach

by Pedro Tradacete

Published in 'La Gaceta de la RSME' Vol. 14, nº 3 (2011) (in Spanish)

We survey several aspects of the invariant subspace problem in Banach spaces: compact operators and Lomonosov theorem,... more We survey several aspects of the invariant subspace problem in Banach spaces: compact operators and Lomonosov theorem, positive operators on Banach lattices, invertible operators and Wermer's theorem, counterexamples and open problems.

Domination problems for strictly singular operators and other related classes

by Pedro Tradacete

To appear in 'Positivity'. In memory of C. D. Aliprantis.

We survey recent results on domination properties of strictly singular operators and related operator ideals, as well... more We survey recent results on domination properties of strictly singular operators and related operator ideals, as well as Banach-Saks operators, Narrow operators and p-summing operators.